/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 86 Sketch a graph of the function. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 86

Sketch a graph of the function. \(f(x)=\frac{\pi}{2}+\arctan x\)

Short Answer

Expert verified
The graph of the function \(f(x) = \frac{\pi}{2} + \arctan x\) is an upward shifted version of the graph of \(\arctan x\) by \(\frac{\pi}{2}\) units. Its horizontal asymptotes are at \(y = 0\) and \(y = \pi\).

Step by step solution

01

Analyzing the base function

Start by examining the graph of the base function, \(\arctan x\). This function generally resembles an S-shape, with a horizontal asymptote (or limit) as \(x\) approaches negative infinity at \(-\frac{\pi}{2}\) and another as \(x\) approaches positive infinity at \(\frac{\pi}{2}\). Thus, the range of \(\arctan x\) is \(-\frac{\pi}{2}<y<\frac{\pi}{2}\)
02

Understanding the impact of a constant addition

Adding a constant to a function results in a vertical shift of the graph. That is, the entire graph of the function is raised or lowered by the value of that constant. In the function \(\frac{\pi}{2}+\arctan x\), the constant addition is \(\frac{\pi}{2}\), so the graph of the function will shift up by \(\frac{\pi}{2}\) units.
03

Drawing the graph

To plot the function \(\frac{\pi}{2}+\arctan x\), take the graph of \(\arctan x\) and shift it upwards by \(\frac{\pi}{2}\) units. This results in new horizontal asymptotes at \(0\) (as \(x\) approaches negative infinity) and at \(\pi\) (as \(x\) approaches positive infinity). The function increases at a declining rate for negative \(x\) values, rises more steeply around \(x=0\) (where the tangent line to the curve is vertical), and then increases at a declining rate for positive \(x\) values.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arctangent Function
The arctangent function, denoted as \( \arctan x \), is a fundamental trigonometric function. It represents the inverse of the tangent function over its restricted domain. The output of the arctangent function is an angle, usually in radians, between the interval \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
Understanding the graph of \( \arctan x \) is key. The curve generally has an S-shape. It has horizontal asymptotes at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \) as \( x \) approaches negative and positive infinity, respectively.
  • These boundaries indicate that the value of \( \arctan x \) will never actually reach these limits but will approach them as \( x \) moves towards extreme values.
  • The steepest section of the curve appears around \( x=0 \), where the rate of change is highest.
The key takeaway is recognizing the arctangent function鈥檚 behavior and the limit it imposes on the output values.
Vertical Shift
The concept of a vertical shift occurs when a constant is added or subtracted from a function. This action translates the entire graph vertically without altering its shape.
The function \( f(x) = \frac{\pi}{2} + \arctan x \) demonstrates a vertical shift. Here, adding \( \frac{\pi}{2} \) results in moving the graph of \( \arctan x \) upwards by this amount.
  • Every point on the graph of \( \arctan x \) shifts up by \( \frac{\pi}{2} \).
  • This means a previously horizontal asymptote of \( y = -\frac{\pi}{2} \) becomes \( y = 0 \).
  • The asymptote at \( y = \frac{\pi}{2} \) shifts to \( y = \pi \).
Such vertical shifts are simple yet highly effective in modifying the overall position of the graph without impacting the core properties of the function.
Horizontal Asymptotes
Horizontal asymptotes are lines that the graph of a function approaches but never touches as \( x \) tends toward infinity or negative infinity. For the arctangent function \( \arctan x \), the horizontal asymptotes occur at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \).
When the vertical shift is applied, as seen in \( f(x) = \frac{\pi}{2} + \arctan x \), these horizontal asymptotes also shift.
  • With an upward shift of \( \frac{\pi}{2} \), the asymptote \( y = -\frac{\pi}{2} \) moves to \( y = 0 \).
  • At the same time, the \( y = \frac{\pi}{2} \) asymptote shifts to \( y = \pi \).
Understanding the movement of horizontal asymptotes during transformations like vertical shifts is essential. It shows how these tweaks can modify the range the function covers over its domain.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the exact value of the expression. (Hint: Sketch a right triangle.) \(\tan \left[\arcsin \left(-\frac{3}{4}\right)\right]\)

Use a calculator to evaluate the expression. Round your result to two decimal places. \(\arctan 2.8\)

Write an algebraic expression that is equivalent to the given expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7.)\) \(\csc \left(\arctan \frac{x}{\sqrt{2}}\right)\)

Write the function in terms of the sine function by using the identity \(A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right).\) Use a graphing utility to graph both forms of the function. What does the graph imply? \(f(t)=4 \cos \pi t+3 \sin \pi t\)

Use an inverse trigonometric function to write \(\theta\) as a function of \(x .\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.